3.4 Complex Zeros and the Fundamental Theorem of Algebra

Size: px
Start display at page:

Download "3.4 Complex Zeros and the Fundamental Theorem of Algebra"

Transcription

1 86 Polynomial Functions.4 Complex Zeros and the Fundamental Theorem of Algebra In Section., we were focused on finding the real zeros of a polynomial function. In this section, we expand our horizons and look for the non-real zeros as well. Consider the polynomial px) = x +1. The zeros of p are the solutions to x + 1 = 0, or x = 1. This equation has no real solutions, but you may recall from Intermediate Algebra that we can formally extract the square roots of both sides to get x = ± 1. The quantity 1 is usually re-labeled i, the so-called imaginary unit. 1 The number i, while not a real number, plays along well with real numbers, and acts very much like any other radical expression. For instance, i) = 6i, 7i i = 4i, 7i) + + 4i) = 5 i, and so forth. The key properties which distinguish i from the real numbers are listed below. Definition.4. The imaginary unit i satisfies the two following properties 1. i = 1. If c is a real number with c 0 then c = i c Property 1 in Definition.4 establishes that i does act as a square root of 1, and property establishes what we mean by the principal square root of a negative real number. In property, it is important to remember the restriction on c. For example, it is perfectly acceptable to say 4 = i 4 = i) = i. However, 4) i 4, otherwise, we d get = 4 = 4) = i 4 = ii) = i = 1) =, which is unacceptable. We are now in the position to define the complex numbers. Definition.5. A complex number is a number of the form a + bi, where a and b are real numbers and i is the imaginary unit. Complex numbers include things you d normally expect, like + i and 5 i. However, don t forget that a or b could be zero, which means numbers like i and 6 are also complex numbers. In other words, don t forget that the complex numbers include the real numbers, so 0 and π 1 are both considered complex numbers. 4 The arithmetic of complex numbers is as you would expect. The only things you need to remember are the two properties in Definition.4. The next example should help recall how these animals behave. 1 Some Technical Mathematics textbooks label it j. Note the use of the indefinite article a. Whatever beast is chosen to be i, i is the other square root of 1. We want to enlarge the number system so we can solve things like x = 1, but not at the cost of the established rules already set in place. For that reason, the general properties of radicals simply do not apply for even roots of negative quantities. 4 See the remarks in Section

2 .4 Complex Zeros and the Fundamental Theorem of Algebra 87 Example.4.1. Perform the indicated operations. Write your answer in the form 5 a + bi i) + 4i). 1 i) + 4i). 1 i 4i ) 1) 6. x [1 + i])x [1 i]) Solution. 1. As mentioned earlier, we treat expressions involving i as we would any other radical. We combine like terms to get 1 i) + 4i) = 1 i 4i = 6i.. Using the distributive property, we get 1 i) + 4i) = 1)) + 1)4i) i)) i)4i) = + 4i 6i 8i. Since i = 1, we get + 4i 6i 8i = i 8) = 11 i.. How in the world are we supposed to simplify 1 i 4i? Well, we deal with the denominator 4i as we would any other denominator containing a radical, and multiply both numerator and denominator by + 4i the conjugate of 4i). 6 Doing so produces 1 i 4i + 4i 1 i) + 4i) 11 i = = = i 4i) + 4i) i 4. We use property of Definition.4 first, then apply the rules of radicals applicable to real radicals to get 1 = i ) i 1 ) = i 1 = 6 = We adhere to the order of operations here and perform the multiplication before the radical to get ) 1) = 6 = We can brute force multiply using the distributive property and see that x [1 + i])x [1 i]) = x x[1 i] x[1 + i] + [1 i][1 + i] = x x + ix x ix + 1 i + i 4i = x x + 5 A couple of remarks about the last example are in order. First, the conjugate of a complex number a + bi is the number a bi. The notation commonly used for conjugation is a bar : a + bi = a bi. For example, + i = i, i = +i, 6 = 6, 4i = 4i, and + 5 = + 5. The properties of the conjugate are summarized in the following theorem. 5 OK, we ll accept things like i even though it can be written as + )i. 6 We will talk more about this in a moment.

3 88 Polynomial Functions Theorem.1. Properties of the Complex Conjugate: Let z and w be complex numbers. z = z z + w = z + w z w = zw z) n = z n, for any natural number n z is a real number if and only if z = z. Essentially, Theorem.1 says that complex conjugation works well with addition, multiplication and powers. The proof of these properties can best be achieved by writing out z = a + bi and w = c + di for real numbers a, b, c and d. Next, we compute the left and right hand sides of each equation and check to see that they are the same. The proof of the first property is a very quick exercise. 7 To prove the second property, we compare z + w and z + w. We have z + w = a + bi + c + di = a bi + c di. To find z + w, we first compute so z + w = a + bi) + c + di) = a + c) + b + d)i z + w = a + c) + b + d)i = a + c) b + d)i = a bi + c di As such, we have established z+w = z + w. The proof for multiplication works similarly. The proof that the conjugate works well with powers can be viewed as a repeated application of the product rule, and is best proved using a technique called Mathematical Induction. 8 The last property is a characterization of real numbers. If z is real, then z = a + 0i, so z = a 0i = a = z. On the other hand, if z = z, then a + bi = a bi which means b = b so b = 0. Hence, z = a + 0i = a and is real. We now return to the business of zeros. Suppose we wish to find the zeros of fx) = x x + 5. To solve the equation x x + 5 = 0, we note that the quadratic doesn t factor nicely, so we resort to the Quadratic Formula, Equation.5 and obtain x = ) ± ) 41)5) 1) = ± 16 = ± 4i = 1 ± i. Two things are important to note. First, the zeros 1 + i and 1 i are complex conjugates. If ever we obtain non-real zeros to a quadratic function with real coefficients, the zeros will be a complex conjugate pair. Do you see why?) Next, we note that in Example.4.1, part 6, we found x [1 + i])x [1 i]) = x x + 5. This demonstrates that the factor theorem holds even for non-real zeros, i.e, x = 1 + i is a zero of f, and, sure enough, x [1 + i]) is a factor of fx). It turns out that polynomial division works the same way for all complex numbers, real and non-real alike, so the Factor and Remainder Theorems hold as well. But how do we know if a 7 Trust us on this. 8 See Section 9..

4 .4 Complex Zeros and the Fundamental Theorem of Algebra 89 general polynomial has any complex zeros at all? We have many examples of polynomials with no real zeros. Can there be polynomials with no zeros whatsoever? The answer to that last question is No. and the theorem which provides that answer is The Fundamental Theorem of Algebra. Theorem.1. The Fundamental Theorem of Algebra: Suppose f is a polynomial function with complex number coefficients of degree n 1, then f has at least one complex zero. The Fundamental Theorem of Algebra is an example of an existence theorem in Mathematics. Like the Intermediate Value Theorem, Theorem.1, the Fundamental Theorem of Algebra guarantees the existence of at least one zero, but gives us no algorithm to use in finding it. In fact, as we mentioned in Section., there are polynomials whose real zeros, though they exist, cannot be expressed using the usual combinations of arithmetic symbols, and must be approximated. The authors are fully aware that the full impact and profound nature of the Fundamental Theorem of Algebra is lost on most students this level, and that s fine. It took mathematicians literally hundreds of years to prove the theorem in its full generality, and some of that history is recorded here. Note that the Fundamental Theorem of Algebra applies to not only polynomial functions with real coefficients, but to those with complex number coefficients as well. Suppose f is a polynomial of degree n 1. The Fundamental Theorem of Algebra guarantees us at least one complex zero, z 1, and as such, the Factor Theorem guarantees that fx) factors as fx) = x z 1 ) q 1 x) for a polynomial function q 1, of degree exactly n 1. If n 1 1, then the Fundamental Theorem of Algebra guarantees a complex zero of q 1 as well, say z, so then the Factor Theorem gives us q 1 x) = x z ) q x), and hence fx) = x z 1 ) x z ) q x). We can continue this process exactly n times, at which point our quotient polynomial q n has degree 0 so it s a constant. This argument gives us the following factorization theorem. Theorem.14. Complex Factorization Theorem: Suppose f is a polynomial function with complex number coefficients. If the degree of f is n and n 1, then f has exactly n complex zeros, counting multiplicity. If z 1, z,..., z k are the distinct zeros of f, with multiplicities m 1, m,..., m k, respectively, then fx) = a x z 1 ) m 1 x z ) m x z k ) m k. Note that the value a in Theorem.14 is the leading coefficient of fx) Can you see why?) and as such, we see that a polynomial is completely determined by its zeros, their multiplicities, and its leading coefficient. We put this theorem to good use in the next example. Example.4.. Let fx) = 1x 5 0x x 6x x Find all of the complex zeros of f and state their multiplicities.. Factor fx) using Theorem.14 Solution. 1. Since f is a fifth degree polynomial, we know that we need to perform at least three successful divisions to get the quotient down to a quadratic function. At that point, we can find the remaining zeros using the Quadratic Formula, if necessary. Using the techniques developed in Section., we get

5 90 Polynomial Functions Our quotient is 1x 1x + 1, whose zeros we find to be 1±i. From Theorem.14, we know f has exactly 5 zeros, counting multiplicities, and as such we have the zero 1 with multiplicity, and the zeros 1, 1+i and 1 i, each of multiplicity 1.. Applying Theorem.14, we are guaranteed that f factors as fx) = 1 x 1 ) x + 1 ) [ x 1 + i ]) [ x 1 i ]) A true test of Theorem.14 and a student s mettle!) would be to take the factored form of fx) in the previous example and multiply it out 9 to see that it really does reduce to the original formula fx) = 1x 5 0x 4 +19x 6x x+1. When factoring a polynomial using Theorem.14, we say that it is factored completely over the complex numbers, meaning that it is impossible to factor the polynomial any further using complex numbers. If we wanted to completely factor fx) over the real numbers then we would have stopped short of finding the nonreal zeros of f and factored f using our work from the synthetic division to write fx) = x ) 1 x + 1 ) 1x 1x + 1 ), or fx) = 1 x 1 ) x + 1 ) x x + 1 ). Since the zeros of x x + 1 are nonreal, we call x x + 1 an irreducible quadratic meaning it is impossible to break it down any further using real numbers. The last two results of the section show us that, at least in theory, if we have a polynomial function with real coefficients, we can always factor it down enough so that any nonreal zeros come from irreducible quadratics. Theorem.15. Conjugate Pairs Theorem: If f is a polynomial function with real number coefficients and z is a zero of f, then so is z. To prove the theorem, suppose f is a polynomial with real number coefficients. Specifically, let fx) = a n x n + a n 1 x n a x + a 1 x + a 0. If z is a zero of f, then fz) = 0, which means a n z n + a n 1 z n a z + a 1 z + a 0 = 0. Next, we consider f z) and apply Theorem.1 below. 9 You really should do this once in your life to convince yourself that all of the theory actually does work!

6 .4 Complex Zeros and the Fundamental Theorem of Algebra 91 f z) = a n z) n + a n 1 z) n a z) + a 1 z + a 0 = a n z n + a n 1 z n a z + a 1 z + a 0 since z) n = z n = a n z n + a n 1 z n a z + a 1 z + a 0 since the coefficients are real = a n z n + a n 1 z n a z + a 1 z + a 0 since z w = zw = a n z n + a n 1 z n a z + a 1 z + a 0 since z + w = z + w = fz) = 0 = 0 This shows that z is a zero of f. So, if f is a polynomial function with real number coefficients, Theorem.15 tells us that if a + bi is a nonreal zero of f, then so is a bi. In other words, nonreal zeros of f come in conjugate pairs. The Factor Theorem kicks in to give us both x [a + bi]) and x [a bi]) as factors of fx) which means x [a + bi])x [a bi]) = x + ax + a + b ) is an irreducible quadratic factor of f. As a result, we have our last theorem of the section. Theorem.16. Real Factorization Theorem: Suppose f is a polynomial function with real number coefficients. Then fx) can be factored into a product of linear factors corresponding to the real zeros of f and irreducible quadratic factors which give the nonreal zeros of f. We now present an example which pulls together all of the major ideas of this section. Example.4.. Let fx) = x Use synthetic division to show that x = + i is a zero of f.. Find the remaining complex zeros of f.. Completely factor fx) over the complex numbers. 4. Completely factor fx) over the real numbers. Solution. 1. Remembering to insert the 0 s in the synthetic division tableau we have + i i 8i i i 8i i 0. Since f is a fourth degree polynomial, we need to make two successful divisions to get a quadratic quotient. Since + i is a zero, we know from Theorem.15 that i is also a zero. We continue our synthetic division tableau.

7 9 Polynomial Functions + i i 8i i 64 i 1 + i 8i i 0 i 8 8i 16 16i Our quotient polynomial is x +4x+8. Using the quadratic formula, we obtain the remaining zeros + i and i.. Using Theorem.14, we get fx) = x [ i])x [ + i])x [ + i])x [ i]). 4. We multiply the linear factors of fx) which correspond to complex conjugate pairs. We find x [ i])x [ + i]) = x 4x + 8, and x [ + i])x [ i]) = x + 4x + 8. Our final answer is fx) = x 4x + 8 ) x + 4x + 8 ). Our last example turns the tables and asks us to manufacture a polynomial with certain properties of its graph and zeros. Example.4.4. Find a polynomial p of lowest degree that has integer coefficients and satisfies all of the following criteria: the graph of y = px) touches but doesn t cross) the x-axis at 1, 0) x = i is a zero of p. as x, px) as x, px) Solution. To solve this problem, we will need a good understanding of the relationship between the x-intercepts of the graph of a function and the zeros of a function, the Factor Theorem, the role of multiplicity, complex conjugates, the Complex Factorization Theorem, and end behavior of polynomial functions. In short, you ll need most of the major concepts of this chapter.) Since the graph of p touches the x-axis at 1, 0), we know x = 1 is a zero of even multiplicity. Since we to have multiplicity exactly. The Factor are after a polynomial of lowest degree, we need x = 1 Theorem now tells us x 1 ) is a factor of px). Since x = i is a zero and our final answer is to have integer real) coefficients, x = i is also a zero. The Factor Theorem kicks in again to give us x i) and x+i) as factors of px). We are given no further information about zeros or intercepts so we conclude, by the Complex Factorization Theorem that px) = a x ) 1 x i)x + i) for some real number a. Expanding this, we get px) = ax 4 a x + 8a 9 x 6ax+a. In order to obtain integer coefficients, we know a must be an integer multiple of 9. Our last concern is end behavior. Since the leading term of px) is ax 4, we need a < 0 to get px) as x ±. Hence, if we choose x = 9, we get px) = 9x 4 + 6x 8x + 54x 9. We can verify our handiwork using the techniques developed in this chapter.

8 .4 Complex Zeros and the Fundamental Theorem of Algebra 9 This example concludes our study of polynomial functions. 10 The last few sections have contained what is considered by many to be heavy Mathematics. Like a heavy meal, heavy Mathematics takes time to digest. Don t be overly concerned if it doesn t seem to sink in all at once, and pace yourself in the Exercises or you re liable to get mental cramps. But before we get to the Exercises, we d like to offer a bit of an epilogue. Our main goal in presenting the material on the complex zeros of a polynomial was to give the chapter a sense of completeness. Given that it can be shown that some polynomials have real zeros which cannot be expressed using the usual algebraic operations, and still others have no real zeros at all, it was nice to discover that every polynomial of degree n 1 has n complex zeros. So like we said, it gives us a sense of closure. But the observant reader will note that we did not give any examples of applications which involve complex numbers. Students often wonder when complex numbers will be used in real-world applications. After all, didn t we call i the imaginary unit? How can imaginary things be used in reality? It turns out that complex numbers are very useful in many applied fields such as fluid dynamics, electromagnetism and quantum mechanics, but most of the applications require Mathematics well beyond College Algebra to fully understand them. That does not mean you ll never be be able to understand them; in fact, it is the authors sincere hope that all of you will reach a point in your studies when the glory, awe and splendor of complex numbers are revealed to you. For now, however, the really good stuff is beyond the scope of this text. We invite you and your classmates to find a few examples of complex number applications and see what you can make of them. A simple Internet search with the phrase complex numbers in real life should get you started. Basic electronics classes are another place to look, but remember, they might use the letter j where we have used i. For the remainder of the text, with the exception of Section 11.7 and a few exploratory exercises scattered about, we will restrict our attention to real numbers. We do this primarily because the first Calculus sequence you will take, ostensibly the one that this text is preparing you for, studies only functions of real variables. Also, lots of really cool scientific things don t require any deep understanding of complex numbers to study them, but they do need more Mathematics like exponential, logarithmic and trigonometric functions. We believe it makes more sense pedagogically for you to learn about those functions now then take a course in Complex Function Theory in your junior or senior year once you ve completed the Calculus sequence. It is in that course that the true power of the complex numbers is released. But for now, in order to fully prepare you for life immediately after College Algebra, we will say that functions like fx) = 1 have a domain of all x +1 real numbers, even though we know x + 1 = 0 has two complex solutions, namely x = ±i. Because x > 0 for all real numbers x, the fraction is never undefined in the real variable setting. x With the exception of the Exercises on the next page, of course.

9 94 Polynomial Functions.4.1 Exercises In Exercises 1-10, use the given complex numbers z and w to find and simplify the following. Write your answers in the form a + bi. z + w zw z 1 z z w w z z zz z) 1. z = + i, w = 4i. z = 1 + i, w = i. z = i, w = 1 + i 4. z = 4i, w = i 5. z = 5i, w = + 7i 6. z = 5 + i, w = 4 + i 7. z = i, w = + i 8. z = 1 i, w = 1 i 9. z = 1 + i, w = 1 + i 10. z = + i, w = i In Exercises 11-18, simplify the quantity ) 4) ) 16) 17. 9) 18. 9) We know that i = 1 which means i = i i = 1) i = i and i 4 = i i = 1) 1) = 1. In Exercises 19-6, use this information to simplify the given power of i. 19. i 5 0. i 6 1. i 7. i 8. i i 6 5. i i 04 In Exercises 7-48, find all of the zeros of the polynomial then completely factor it over the real numbers and completely factor it over the complex numbers. 7. fx) = x 4x fx) = x x fx) = x + x fx) = x x + 9x fx) = x + 6x + 6x + 5. fx) = x 1x + 4x 1

10 .4 Complex Zeros and the Fundamental Theorem of Algebra 95. fx) = x + x + 4x fx) = 4x 6x 8x fx) = x + 7x + 9x 6. fx) = 9x + x fx) = 4x 4 4x + 1x 1x + 8. fx) = x 4 7x + 14x 15x fx) = x 4 + x + 7x + 9x fx) = 6x x 55x + 16x fx) = x 4 8x 1x 1x 5 4. fx) = 8x x + 4x + x 4 4. fx) = x 4 + 9x fx) = x 4 + 5x fx) = x 5 x 4 + 7x 7x + 1x fx) = x fx) = x 4 x + 7x x + 6 Hint: x = i is one of the zeros.) 48. fx) = x 4 + 5x + 1x + 7x + 5 Hint: x = 1 + i is a zero.) In Exercises 49-5, create a polynomial f with real number coefficients which has all of the desired characteristics. You may leave the polynomial in factored form. 49. The zeros of f are c = ±1 and c = ±i The leading term of fx) is 4x c = i is a zero. the point 1, 0) is a local minimum on the graph of y = fx) the leading term of fx) is 117x The solutions to fx) = 0 are x = ± and x = ±7i The leading term of fx) is x 5 The point, 0) is a local maximum on the graph of y = fx). 5. f is degree 5. x = 6, x = i and x = 1 i are zeros of f as x, fx) 5. The leading term of fx) is x c = i is a zero f0) = Let z and w be arbitrary complex numbers. Show that z w = zw and z = z.

11 96 Polynomial Functions.4. Answers 1. For z = + i and w = 4i z + w = + 7i zw = 1 + 8i z = 5 + 1i 1 z = 1 1 i z w = 4 1 i w z = i z = i zz = 1 z) = 5 1i. For z = 1 + i and w = i z + w = 1 zw = 1 i z = i 1 z = 1 1 i z w = 1 + i w z = 1 1 i z = 1 i zz = z) = i. For z = i and w = 1 + i z + w = 1 + i zw = i z = 1 1 z = i z w = i w z = + i z = i zz = 1 z) = 1 4. For z = 4i and w = i z + w = + i zw = 8 + 8i z = 16 1 z = 1 4 i z w = 1 + i w z = 1 1 i z = 4i zz = 16 z) = For z = 5i and w = + 7i z + w = 5 + i zw = i z = 16 0i 1 z = i z w = i w z = i z = + 5i zz = 4 z) = i

12 .4 Complex Zeros and the Fundamental Theorem of Algebra For z = 5 + i and w = 4 + i z + w = 1 + i zw = 6i z = 4 10i 1 z = i z w = i w z = i z = 5 i zz = 6 z) = i 7. For z = i and w = + i z + w = zw = 4 z = 4i 1 z = i z w = i w z = i z = + i zz = 4 z) = 4i 8. For z = 1 i and w = 1 i z + w = i zw = 4 z = i 1 z = i z w = 1 + i w z = 1 i z = 1 + i zz = 4 z) = + i 9. For z = 1 + i and w = 1 + i z + w = i zw = 1 z = 1 + i 1 z = 1 i z w = 1 i w z = 1 + i z = 1 i zz = 1 z) = 1 i 10. For z = + i and w = i zw = 1 z = i 1 z = i z w = i w z = i z = i zz = 1 z) = i 11. 7i 1. i

13 98 Polynomial Functions i 19. i 5 = i 4 i = 1 i = i 0. i 6 = i 4 i = 1 1) = 1 1. i 7 = i 4 i = 1 i) = i. i 8 = i 4 i 4 = i 4) = 1) = 1. i 15 = i 4) i = 1 i) = i 4. i 6 = i 4) 6 i = 1 1) = 1 5. i 117 = i 4) 9 i = 1 i = i 6. i 04 = i 4) 76 = 1 76 = 1 7. fx) = x 4x + 1 = x + i))x i)) Zeros: x = ± i 8. fx) = x x + 5 = x 1 + i))x 1 i)) Zeros: x = 1 ± i 9. fx) = x + x + 10 = x 1 )) + 9 x i 1 )) 9 i Zeros: x = 1 ± 9 i 0. fx) = x x + 9x 18 = x ) x + 9 ) = x )x i)x + i) Zeros: x =, ±i 1. fx) = x +6x +6x+5 = x+5)x +x+1) = x+5) x 1 )) + x i 1 )) i Zeros: x = 5, x = 1 ± i. fx) = x 1x + 4x 1 = x 1)x 4x + 1) = x 1)x + i))x i)) Zeros: x = 1, x = ± i. fx) = x + x + 4x + 1 = x + ) x + 4 ) = x + )x + i)x i) Zeros: x =, ±i 4. fx) = 4x 6x 8x + 15 = x + ) 4x 1x + 10 ) = 4 x + ) x + 1 i)) x 1 i)) Zeros: x =, x = ± 1 i 5. fx) = x + 7x + 9x = x + ) x 5 )) + 9 x 5 9 Zeros: x =, x = 5 ± 9 6. fx) = 9x + x + 1 = x + 1 ) 9x x + ) = 9 x + 1 ) x Zeros: x = 1, x = 1 6 ± 11 6 i i )) x i )) 7. fx) = 4x 4 4x + 1x 1x + = x 1 ) 4x + 1 ) = 4 x 1 ) x + i )x i ) Zeros: x = 1, x = ± i ))

14 .4 Complex Zeros and the Fundamental Theorem of Algebra fx) = x 4 7x + 14x 15x + 6 = x 1) x x + 6 ) = x 1) x )) )) x i i Zeros: x = 1, x = 4 ± 9 4 i 9. fx) = x 4 + x + 7x + 9x 18 = x + )x 1) x + 9 ) = x + )x 1)x + i)x i) Zeros: x =, 1, ±i 40. fx) = 6x 4 +17x 55x +16x+1 = 6 x + ) 1 ) )) )) x x + x Zeros: x = 1, x =, x = ± 41. fx) = x 4 8x 1x 1x 5 = x + 1) x x 5 ) = x + 1) x 1 )) + 14 x i 1 )) 14 i Zeros: x = 1, x = 1 ± 14 i 4. fx) = 8x x + 4x + x 4 = 8 x + 1 ) x 1 4) x + 5))x 5)) Zeros: x = 1, 1 4, x = ± 5 4. fx) = x 4 + 9x + 0 = x + 4 ) x + 5 ) = x i)x + i) x i 5 ) x + i 5 ) Zeros: x = ±i, ±i fx) = x 4 + 5x 4 = x ) x + 8 ) = x )x + ) x i ) x + i ) Zeros: x = ±, ±i 45. fx) = x 5 x 4 + 7x 7x + 1x 1 = x 1) x + ) x + 4 ) = x 1)x i )x + i )x i)x + i) Zeros: x = 1, ± i, ±i 46. fx) = x 6 64 = x )x + ) x + x + 4 ) x x + 4 ) = x )x + ) x 1 + i )) x 1 i )) x 1 + i )) x 1 i )) Zeros: x = ±, x = 1 ± i, x = 1 ± i 47. fx) = x 4 x +7x x+6 = x x+6)x +1) = x 1+5i))x 1 5i))x+i)x i) Zeros: x = 1 ± 5i, x = ±i 48. fx) = x 4 + 5x + 1x + 7x + 5 = x + x + 5 ) x + x + 1 ) = x 1 + i))x 1 i)) Zeros: x = 1 ± i, 1 4 ± i 7 4 x i 7 4 )) x 1 4 i fx) = 4x 1)x + 1)x i)x + i) 50. fx) = 117x + 1) x i)x + i) 51. fx) = x ) x + )x 7i)x + 7i) 5. fx) = ax 6)x i)x + i)x 1 i))x 1 + i)) where a is any real number, a < 0 5. fx) = x i)x + i)x + ) ))

Zeros of a Polynomial Function

Zeros of a Polynomial Function Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we

More information

Zeros of Polynomial Functions

Zeros of Polynomial Functions Review: Synthetic Division Find (x 2-5x - 5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 3-5x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 3-5x 2 + x + 2. Zeros of Polynomial Functions Introduction

More information

3.2 The Factor Theorem and The Remainder Theorem

3.2 The Factor Theorem and The Remainder Theorem 3. The Factor Theorem and The Remainder Theorem 57 3. The Factor Theorem and The Remainder Theorem Suppose we wish to find the zeros of f(x) = x 3 + 4x 5x 4. Setting f(x) = 0 results in the polynomial

More information

Zeros of Polynomial Functions

Zeros of Polynomial Functions Zeros of Polynomial Functions Objectives: 1.Use the Fundamental Theorem of Algebra to determine the number of zeros of polynomial functions 2.Find rational zeros of polynomial functions 3.Find conjugate

More information

2.3. Finding polynomial functions. An Introduction:

2.3. Finding polynomial functions. An Introduction: 2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned

More information

0.8 Rational Expressions and Equations

0.8 Rational Expressions and Equations 96 Prerequisites 0.8 Rational Expressions and Equations We now turn our attention to rational expressions - that is, algebraic fractions - and equations which contain them. The reader is encouraged to

More information

Zeros of Polynomial Functions

Zeros of Polynomial Functions Zeros of Polynomial Functions The Rational Zero Theorem If f (x) = a n x n + a n-1 x n-1 + + a 1 x + a 0 has integer coefficients and p/q (where p/q is reduced) is a rational zero, then p is a factor of

More information

Answer Key for California State Standards: Algebra I

Answer Key for California State Standards: Algebra I Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.

More information

Differentiation and Integration

Differentiation and Integration This material is a supplement to Appendix G of Stewart. You should read the appendix, except the last section on complex exponentials, before this material. Differentiation and Integration Suppose we have

More information

5.1 Radical Notation and Rational Exponents

5.1 Radical Notation and Rational Exponents Section 5.1 Radical Notation and Rational Exponents 1 5.1 Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots

More information

Polynomial and Rational Functions

Polynomial and Rational Functions Polynomial and Rational Functions Quadratic Functions Overview of Objectives, students should be able to: 1. Recognize the characteristics of parabolas. 2. Find the intercepts a. x intercepts by solving

More information

March 29, 2011. 171S4.4 Theorems about Zeros of Polynomial Functions

March 29, 2011. 171S4.4 Theorems about Zeros of Polynomial Functions MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial

More information

Zero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.

Zero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P. MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called

More information

3.3 Real Zeros of Polynomials

3.3 Real Zeros of Polynomials 3.3 Real Zeros of Polynomials 69 3.3 Real Zeros of Polynomials In Section 3., we found that we can use synthetic division to determine if a given real number is a zero of a polynomial function. This section

More information

Polynomials. Dr. philippe B. laval Kennesaw State University. April 3, 2005

Polynomials. Dr. philippe B. laval Kennesaw State University. April 3, 2005 Polynomials Dr. philippe B. laval Kennesaw State University April 3, 2005 Abstract Handout on polynomials. The following topics are covered: Polynomial Functions End behavior Extrema Polynomial Division

More information

9. POLYNOMIALS. Example 1: The expression a(x) = x 3 4x 2 + 7x 11 is a polynomial in x. The coefficients of a(x) are the numbers 1, 4, 7, 11.

9. POLYNOMIALS. Example 1: The expression a(x) = x 3 4x 2 + 7x 11 is a polynomial in x. The coefficients of a(x) are the numbers 1, 4, 7, 11. 9. POLYNOMIALS 9.1. Definition of a Polynomial A polynomial is an expression of the form: a(x) = a n x n + a n-1 x n-1 +... + a 1 x + a 0. The symbol x is called an indeterminate and simply plays the role

More information

Section 1.1 Linear Equations: Slope and Equations of Lines

Section 1.1 Linear Equations: Slope and Equations of Lines Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of

More information

Solving Quadratic Equations

Solving Quadratic Equations 9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation

More information

The Method of Partial Fractions Math 121 Calculus II Spring 2015

The Method of Partial Fractions Math 121 Calculus II Spring 2015 Rational functions. as The Method of Partial Fractions Math 11 Calculus II Spring 015 Recall that a rational function is a quotient of two polynomials such f(x) g(x) = 3x5 + x 3 + 16x x 60. The method

More information

Week 13 Trigonometric Form of Complex Numbers

Week 13 Trigonometric Form of Complex Numbers Week Trigonometric Form of Complex Numbers Overview In this week of the course, which is the last week if you are not going to take calculus, we will look at how Trigonometry can sometimes help in working

More information

SOLVING POLYNOMIAL EQUATIONS

SOLVING POLYNOMIAL EQUATIONS C SOLVING POLYNOMIAL EQUATIONS We will assume in this appendix that you know how to divide polynomials using long division and synthetic division. If you need to review those techniques, refer to an algebra

More information

MA107 Precalculus Algebra Exam 2 Review Solutions

MA107 Precalculus Algebra Exam 2 Review Solutions MA107 Precalculus Algebra Exam 2 Review Solutions February 24, 2008 1. The following demand equation models the number of units sold, x, of a product as a function of price, p. x = 4p + 200 a. Please write

More information

JUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson

JUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson JUST THE MATHS UNIT NUMBER 1.8 ALGEBRA 8 (Polynomials) by A.J.Hobson 1.8.1 The factor theorem 1.8.2 Application to quadratic and cubic expressions 1.8.3 Cubic equations 1.8.4 Long division of polynomials

More information

1.6 The Order of Operations

1.6 The Order of Operations 1.6 The Order of Operations Contents: Operations Grouping Symbols The Order of Operations Exponents and Negative Numbers Negative Square Roots Square Root of a Negative Number Order of Operations and Negative

More information

Chapter 7 - Roots, Radicals, and Complex Numbers

Chapter 7 - Roots, Radicals, and Complex Numbers Math 233 - Spring 2009 Chapter 7 - Roots, Radicals, and Complex Numbers 7.1 Roots and Radicals 7.1.1 Notation and Terminology In the expression x the is called the radical sign. The expression under the

More information

MATH 10034 Fundamental Mathematics IV

MATH 10034 Fundamental Mathematics IV MATH 0034 Fundamental Mathematics IV http://www.math.kent.edu/ebooks/0034/funmath4.pdf Department of Mathematical Sciences Kent State University January 2, 2009 ii Contents To the Instructor v Polynomials.

More information

6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives

6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives 6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise

More information

SECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS

SECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS SECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS Assume f ( x) is a nonconstant polynomial with real coefficients written in standard form. PART A: TECHNIQUES WE HAVE ALREADY SEEN Refer to: Notes 1.31

More information

Partial Fractions. (x 1)(x 2 + 1)

Partial Fractions. (x 1)(x 2 + 1) Partial Fractions Adding rational functions involves finding a common denominator, rewriting each fraction so that it has that denominator, then adding. For example, 3x x 1 3x(x 1) (x + 1)(x 1) + 1(x +

More information

College Algebra - MAT 161 Page: 1 Copyright 2009 Killoran

College Algebra - MAT 161 Page: 1 Copyright 2009 Killoran College Algebra - MAT 6 Page: Copyright 2009 Killoran Zeros and Roots of Polynomial Functions Finding a Root (zero or x-intercept) of a polynomial is identical to the process of factoring a polynomial.

More information

The degree of a polynomial function is equal to the highest exponent found on the independent variables.

The degree of a polynomial function is equal to the highest exponent found on the independent variables. DETAILED SOLUTIONS AND CONCEPTS - POLYNOMIAL FUNCTIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! PLEASE NOTE

More information

Exponents and Radicals

Exponents and Radicals Exponents and Radicals (a + b) 10 Exponents are a very important part of algebra. An exponent is just a convenient way of writing repeated multiplications of the same number. Radicals involve the use of

More information

POLYNOMIAL FUNCTIONS

POLYNOMIAL FUNCTIONS POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a

More information

a 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4)

a 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4) ROOTS OF POLYNOMIAL EQUATIONS In this unit we discuss polynomial equations. A polynomial in x of degree n, where n 0 is an integer, is an expression of the form P n (x) =a n x n + a n 1 x n 1 + + a 1 x

More information

Math 4310 Handout - Quotient Vector Spaces

Math 4310 Handout - Quotient Vector Spaces Math 4310 Handout - Quotient Vector Spaces Dan Collins The textbook defines a subspace of a vector space in Chapter 4, but it avoids ever discussing the notion of a quotient space. This is understandable

More information

Review of Fundamental Mathematics

Review of Fundamental Mathematics Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decision-making tools

More information

1 Lecture: Integration of rational functions by decomposition

1 Lecture: Integration of rational functions by decomposition Lecture: Integration of rational functions by decomposition into partial fractions Recognize and integrate basic rational functions, except when the denominator is a power of an irreducible quadratic.

More information

Application. Outline. 3-1 Polynomial Functions 3-2 Finding Rational Zeros of. Polynomial. 3-3 Approximating Real Zeros of.

Application. Outline. 3-1 Polynomial Functions 3-2 Finding Rational Zeros of. Polynomial. 3-3 Approximating Real Zeros of. Polynomial and Rational Functions Outline 3-1 Polynomial Functions 3-2 Finding Rational Zeros of Polynomials 3-3 Approximating Real Zeros of Polynomials 3-4 Rational Functions Chapter 3 Group Activity:

More information

This unit will lay the groundwork for later units where the students will extend this knowledge to quadratic and exponential functions.

This unit will lay the groundwork for later units where the students will extend this knowledge to quadratic and exponential functions. Algebra I Overview View unit yearlong overview here Many of the concepts presented in Algebra I are progressions of concepts that were introduced in grades 6 through 8. The content presented in this course

More information

3.1. RATIONAL EXPRESSIONS

3.1. RATIONAL EXPRESSIONS 3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers

More information

Indiana State Core Curriculum Standards updated 2009 Algebra I

Indiana State Core Curriculum Standards updated 2009 Algebra I Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and

More information

3.6. The factor theorem

3.6. The factor theorem 3.6. The factor theorem Example 1. At the right we have drawn the graph of the polynomial y = x 4 9x 3 + 8x 36x + 16. Your problem is to write the polynomial in factored form. Does the geometry of the

More information

Chapter 4. Polynomial and Rational Functions. 4.1 Polynomial Functions and Their Graphs

Chapter 4. Polynomial and Rational Functions. 4.1 Polynomial Functions and Their Graphs Chapter 4. Polynomial and Rational Functions 4.1 Polynomial Functions and Their Graphs A polynomial function of degree n is a function of the form P = a n n + a n 1 n 1 + + a 2 2 + a 1 + a 0 Where a s

More information

Vieta s Formulas and the Identity Theorem

Vieta s Formulas and the Identity Theorem Vieta s Formulas and the Identity Theorem This worksheet will work through the material from our class on 3/21/2013 with some examples that should help you with the homework The topic of our discussion

More information

1.7 Graphs of Functions

1.7 Graphs of Functions 64 Relations and Functions 1.7 Graphs of Functions In Section 1.4 we defined a function as a special type of relation; one in which each x-coordinate was matched with only one y-coordinate. We spent most

More information

4.1. COMPLEX NUMBERS

4.1. COMPLEX NUMBERS 4.1. COMPLEX NUMBERS What You Should Learn Use the imaginary unit i to write complex numbers. Add, subtract, and multiply complex numbers. Use complex conjugates to write the quotient of two complex numbers

More information

Solutions of Linear Equations in One Variable

Solutions of Linear Equations in One Variable 2. Solutions of Linear Equations in One Variable 2. OBJECTIVES. Identify a linear equation 2. Combine like terms to solve an equation We begin this chapter by considering one of the most important tools

More information

COMPLEX NUMBERS. a bi c di a c b d i. a bi c di a c b d i For instance, 1 i 4 7i 1 4 1 7 i 5 6i

COMPLEX NUMBERS. a bi c di a c b d i. a bi c di a c b d i For instance, 1 i 4 7i 1 4 1 7 i 5 6i COMPLEX NUMBERS _4+i _-i FIGURE Complex numbers as points in the Arg plane i _i +i -i A complex number can be represented by an expression of the form a bi, where a b are real numbers i is a symbol with

More information

ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form

ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form Goal Graph quadratic functions. VOCABULARY Quadratic function A function that can be written in the standard form y = ax 2 + bx+ c where a 0 Parabola

More information

Partial Fractions. Combining fractions over a common denominator is a familiar operation from algebra:

Partial Fractions. Combining fractions over a common denominator is a familiar operation from algebra: Partial Fractions Combining fractions over a common denominator is a familiar operation from algebra: From the standpoint of integration, the left side of Equation 1 would be much easier to work with than

More information

Algebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 2012-13 school year.

Algebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 2012-13 school year. This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra

More information

Algebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions.

Algebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions. Chapter 1 Vocabulary identity - A statement that equates two equivalent expressions. verbal model- A word equation that represents a real-life problem. algebraic expression - An expression with variables.

More information

Polynomial Degree and Finite Differences

Polynomial Degree and Finite Differences CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial

More information

Definition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality.

Definition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality. 8 Inequalities Concepts: Equivalent Inequalities Linear and Nonlinear Inequalities Absolute Value Inequalities (Sections 4.6 and 1.1) 8.1 Equivalent Inequalities Definition 8.1 Two inequalities are equivalent

More information

Lecture Notes on Polynomials

Lecture Notes on Polynomials Lecture Notes on Polynomials Arne Jensen Department of Mathematical Sciences Aalborg University c 008 Introduction These lecture notes give a very short introduction to polynomials with real and complex

More information

The Point-Slope Form

The Point-Slope Form 7. The Point-Slope Form 7. OBJECTIVES 1. Given a point and a slope, find the graph of a line. Given a point and the slope, find the equation of a line. Given two points, find the equation of a line y Slope

More information

63. Graph y 1 2 x and y 2 THE FACTOR THEOREM. The Factor Theorem. Consider the polynomial function. P(x) x 2 2x 15.

63. Graph y 1 2 x and y 2 THE FACTOR THEOREM. The Factor Theorem. Consider the polynomial function. P(x) x 2 2x 15. 9.4 (9-27) 517 Gear ratio d) For a fixed wheel size and chain ring, does the gear ratio increase or decrease as the number of teeth on the cog increases? decreases 100 80 60 40 20 27-in. wheel, 44 teeth

More information

3.6 The Real Zeros of a Polynomial Function

3.6 The Real Zeros of a Polynomial Function SECTION 3.6 The Real Zeros of a Polynomial Function 219 3.6 The Real Zeros of a Polynomial Function PREPARING FOR THIS SECTION Before getting started, review the following: Classification of Numbers (Appendix,

More information

South Carolina College- and Career-Ready (SCCCR) Algebra 1

South Carolina College- and Career-Ready (SCCCR) Algebra 1 South Carolina College- and Career-Ready (SCCCR) Algebra 1 South Carolina College- and Career-Ready Mathematical Process Standards The South Carolina College- and Career-Ready (SCCCR) Mathematical Process

More information

Examples of Tasks from CCSS Edition Course 3, Unit 5

Examples of Tasks from CCSS Edition Course 3, Unit 5 Examples of Tasks from CCSS Edition Course 3, Unit 5 Getting Started The tasks below are selected with the intent of presenting key ideas and skills. Not every answer is complete, so that teachers can

More information

26 Integers: Multiplication, Division, and Order

26 Integers: Multiplication, Division, and Order 26 Integers: Multiplication, Division, and Order Integer multiplication and division are extensions of whole number multiplication and division. In multiplying and dividing integers, the one new issue

More information

MATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab

MATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab MATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab MATH 0110 is established to accommodate students desiring non-course based remediation in developmental mathematics. This structure will

More information

NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS

NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS TEST DESIGN AND FRAMEWORK September 2014 Authorized for Distribution by the New York State Education Department This test design and framework document

More information

2.5 Zeros of a Polynomial Functions

2.5 Zeros of a Polynomial Functions .5 Zeros of a Polynomial Functions Section.5 Notes Page 1 The first rule we will talk about is Descartes Rule of Signs, which can be used to determine the possible times a graph crosses the x-axis and

More information

Quotient Rings and Field Extensions

Quotient Rings and Field Extensions Chapter 5 Quotient Rings and Field Extensions In this chapter we describe a method for producing field extension of a given field. If F is a field, then a field extension is a field K that contains F.

More information

Properties of Real Numbers

Properties of Real Numbers 16 Chapter P Prerequisites P.2 Properties of Real Numbers What you should learn: Identify and use the basic properties of real numbers Develop and use additional properties of real numbers Why you should

More information

Solving Rational Equations

Solving Rational Equations Lesson M Lesson : Student Outcomes Students solve rational equations, monitoring for the creation of extraneous solutions. Lesson Notes In the preceding lessons, students learned to add, subtract, multiply,

More information

3-17 15-25 5 15-10 25 3-2 5 0. 1b) since the remainder is 0 I need to factor the numerator. Synthetic division tells me this is true

3-17 15-25 5 15-10 25 3-2 5 0. 1b) since the remainder is 0 I need to factor the numerator. Synthetic division tells me this is true Section 5.2 solutions #1-10: a) Perform the division using synthetic division. b) if the remainder is 0 use the result to completely factor the dividend (this is the numerator or the polynomial to the

More information

This is a square root. The number under the radical is 9. (An asterisk * means multiply.)

This is a square root. The number under the radical is 9. (An asterisk * means multiply.) Page of Review of Radical Expressions and Equations Skills involving radicals can be divided into the following groups: Evaluate square roots or higher order roots. Simplify radical expressions. Rationalize

More information

Algebra II End of Course Exam Answer Key Segment I. Scientific Calculator Only

Algebra II End of Course Exam Answer Key Segment I. Scientific Calculator Only Algebra II End of Course Exam Answer Key Segment I Scientific Calculator Only Question 1 Reporting Category: Algebraic Concepts & Procedures Common Core Standard: A-APR.3: Identify zeros of polynomials

More information

Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.

Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers. Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used

More information

LAKE ELSINORE UNIFIED SCHOOL DISTRICT

LAKE ELSINORE UNIFIED SCHOOL DISTRICT LAKE ELSINORE UNIFIED SCHOOL DISTRICT Title: PLATO Algebra 1-Semester 2 Grade Level: 10-12 Department: Mathematics Credit: 5 Prerequisite: Letter grade of F and/or N/C in Algebra 1, Semester 2 Course Description:

More information

parent ROADMAP MATHEMATICS SUPPORTING YOUR CHILD IN HIGH SCHOOL

parent ROADMAP MATHEMATICS SUPPORTING YOUR CHILD IN HIGH SCHOOL parent ROADMAP MATHEMATICS SUPPORTING YOUR CHILD IN HIGH SCHOOL HS America s schools are working to provide higher quality instruction than ever before. The way we taught students in the past simply does

More information

Introduction to Fractions

Introduction to Fractions Section 0.6 Contents: Vocabulary of Fractions A Fraction as division Undefined Values First Rules of Fractions Equivalent Fractions Building Up Fractions VOCABULARY OF FRACTIONS Simplifying Fractions Multiplying

More information

Polynomial and Synthetic Division. Long Division of Polynomials. Example 1. 6x 2 7x 2 x 2) 19x 2 16x 4 6x3 12x 2 7x 2 16x 7x 2 14x. 2x 4.

Polynomial and Synthetic Division. Long Division of Polynomials. Example 1. 6x 2 7x 2 x 2) 19x 2 16x 4 6x3 12x 2 7x 2 16x 7x 2 14x. 2x 4. _.qd /7/5 9: AM Page 5 Section.. Polynomial and Synthetic Division 5 Polynomial and Synthetic Division What you should learn Use long division to divide polynomials by other polynomials. Use synthetic

More information

Lies My Calculator and Computer Told Me

Lies My Calculator and Computer Told Me Lies My Calculator and Computer Told Me 2 LIES MY CALCULATOR AND COMPUTER TOLD ME Lies My Calculator and Computer Told Me See Section.4 for a discussion of graphing calculators and computers with graphing

More information

PRE-CALCULUS GRADE 12

PRE-CALCULUS GRADE 12 PRE-CALCULUS GRADE 12 [C] Communication Trigonometry General Outcome: Develop trigonometric reasoning. A1. Demonstrate an understanding of angles in standard position, expressed in degrees and radians.

More information

ALGEBRA 2 CRA 2 REVIEW - Chapters 1-6 Answer Section

ALGEBRA 2 CRA 2 REVIEW - Chapters 1-6 Answer Section ALGEBRA 2 CRA 2 REVIEW - Chapters 1-6 Answer Section MULTIPLE CHOICE 1. ANS: C 2. ANS: A 3. ANS: A OBJ: 5-3.1 Using Vertex Form SHORT ANSWER 4. ANS: (x + 6)(x 2 6x + 36) OBJ: 6-4.2 Solving Equations by

More information

Trigonometric Functions and Equations

Trigonometric Functions and Equations Contents Trigonometric Functions and Equations Lesson 1 Reasoning with Trigonometric Functions Investigations 1 Proving Trigonometric Identities... 271 2 Sum and Difference Identities... 276 3 Extending

More information

Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.

Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. Algebra 2 - Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers - {1,2,3,4,...}

More information

Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks

Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson

More information

MTH124: Honors Algebra I

MTH124: Honors Algebra I MTH124: Honors Algebra I This course prepares students for more advanced courses while they develop algebraic fluency, learn the skills needed to solve equations, and perform manipulations with numbers,

More information

MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education)

MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education) MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education) Accurately add, subtract, multiply, and divide whole numbers, integers,

More information

Procedure for Graphing Polynomial Functions

Procedure for Graphing Polynomial Functions Procedure for Graphing Polynomial Functions P(x) = a n x n + a n-1 x n-1 + + a 1 x + a 0 To graph P(x): As an example, we will examine the following polynomial function: P(x) = 2x 3 3x 2 23x + 12 1. Determine

More information

1 Shapes of Cubic Functions

1 Shapes of Cubic Functions MA 1165 - Lecture 05 1 1/26/09 1 Shapes of Cubic Functions A cubic function (a.k.a. a third-degree polynomial function) is one that can be written in the form f(x) = ax 3 + bx 2 + cx + d. (1) Quadratic

More information

Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

More information

3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes

3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes Solving Polynomial Equations 3.3 Introduction Linear and quadratic equations, dealt within Sections 3.1 and 3.2, are members of a class of equations, called polynomial equations. These have the general

More information

SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS

SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS (Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.1 SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVES Be able to identify polynomial, rational, and algebraic

More information

Revised Version of Chapter 23. We learned long ago how to solve linear congruences. ax c (mod m)

Revised Version of Chapter 23. We learned long ago how to solve linear congruences. ax c (mod m) Chapter 23 Squares Modulo p Revised Version of Chapter 23 We learned long ago how to solve linear congruences ax c (mod m) (see Chapter 8). It s now time to take the plunge and move on to quadratic equations.

More information

Chapter 11 Number Theory

Chapter 11 Number Theory Chapter 11 Number Theory Number theory is one of the oldest branches of mathematics. For many years people who studied number theory delighted in its pure nature because there were few practical applications

More information

Polynomials and Factoring

Polynomials and Factoring Lesson 2 Polynomials and Factoring A polynomial function is a power function or the sum of two or more power functions, each of which has a nonnegative integer power. Because polynomial functions are built

More information

Algebra 1. Curriculum Map

Algebra 1. Curriculum Map Algebra 1 Curriculum Map Table of Contents Unit 1: Expressions and Unit 2: Linear Unit 3: Representing Linear Unit 4: Linear Inequalities Unit 5: Systems of Linear Unit 6: Polynomials Unit 7: Factoring

More information

A Quick Algebra Review

A Quick Algebra Review 1. Simplifying Epressions. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Eponents 9. Quadratics 10. Rationals 11. Radicals

More information

Vocabulary Words and Definitions for Algebra

Vocabulary Words and Definitions for Algebra Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms

More information

Higher Education Math Placement

Higher Education Math Placement Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication

More information

DRAFT. Algebra 1 EOC Item Specifications

DRAFT. Algebra 1 EOC Item Specifications DRAFT Algebra 1 EOC Item Specifications The draft Florida Standards Assessment (FSA) Test Item Specifications (Specifications) are based upon the Florida Standards and the Florida Course Descriptions as

More information

Georgia Standards of Excellence Curriculum Frameworks. Mathematics. GSE Algebra II/Advanced Algebra Unit 1: Quadratics Revisited

Georgia Standards of Excellence Curriculum Frameworks. Mathematics. GSE Algebra II/Advanced Algebra Unit 1: Quadratics Revisited Georgia Standards of Excellence Curriculum Frameworks Mathematics GSE Algebra II/Advanced Algebra Unit 1: Quadratics Revisited These materials are for nonprofit educational purposes only. Any other use

More information

6.4 Logarithmic Equations and Inequalities

6.4 Logarithmic Equations and Inequalities 6.4 Logarithmic Equations and Inequalities 459 6.4 Logarithmic Equations and Inequalities In Section 6.3 we solved equations and inequalities involving exponential functions using one of two basic strategies.

More information

Polynomial Expressions and Equations

Polynomial Expressions and Equations Polynomial Expressions and Equations This is a really close-up picture of rain. Really. The picture represents falling water broken down into molecules, each with two hydrogen atoms connected to one oxygen

More information

SIMPLIFYING ALGEBRAIC FRACTIONS

SIMPLIFYING ALGEBRAIC FRACTIONS Tallahassee Community College 5 SIMPLIFYING ALGEBRAIC FRACTIONS In arithmetic, you learned that a fraction is in simplest form if the Greatest Common Factor (GCF) of the numerator and the denominator is

More information